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Your data matches 36 different statistics following compositions of up to 3 maps.
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Matching statistic: St000160
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 2
[3]
=> 1
[2,1]
=> 1
[1,1,1]
=> 3
[4]
=> 1
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 2
[1,1,1,1]
=> 4
[5]
=> 1
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 2
[2,2,1]
=> 1
[2,1,1,1]
=> 3
[1,1,1,1,1]
=> 5
[6]
=> 1
[5,1]
=> 1
[4,2]
=> 1
[4,1,1]
=> 2
[3,3]
=> 2
[3,2,1]
=> 1
[3,1,1,1]
=> 3
[2,2,2]
=> 3
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> 6
[7]
=> 1
[6,1]
=> 1
[5,1,1]
=> 2
[4,1,1,1]
=> 3
[3,1,1,1,1]
=> 4
[1,1,1,1,1,1,1]
=> 7
[8]
=> 1
[7,1]
=> 1
[6,1,1]
=> 2
[5,1,1,1]
=> 3
[4,4]
=> 2
[2,2,2,2]
=> 4
[1,1,1,1,1,1,1,1]
=> 8
[9]
=> 1
[8,1]
=> 1
[7,1,1]
=> 2
[1,1,1,1,1,1,1,1,1]
=> 9
[10]
=> 1
[9,1]
=> 1
[2,2,2,2,2]
=> 5
[1,1,1,1,1,1,1,1,1,1]
=> 10
Description
The multiplicity of the smallest part of a partition.
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000153
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> {{1}}
=> [1] => 1
[2]
=> [[1,2]]
=> {{1,2}}
=> [2,1] => 1
[1,1]
=> [[1],[2]]
=> {{1},{2}}
=> [1,2] => 2
[3]
=> [[1,2,3]]
=> {{1,2,3}}
=> [2,3,1] => 1
[2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> [3,2,1] => 1
[1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> [1,2,3] => 3
[4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> [2,3,4,1] => 1
[3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> [3,2,4,1] => 1
[2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> [2,1,4,3] => 2
[2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> [4,2,3,1] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => 4
[5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
[4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => 1
[3,2]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> [4,2,3,5,1] => 2
[2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 5
[6]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> {{1,4,5,6},{2},{3}}
=> [4,2,3,5,6,1] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => 2
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> {{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> {{1,5,6},{2},{3},{4}}
=> [5,2,3,4,6,1] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => 3
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> {{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => 2
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> {{1,6},{2},{3},{4},{5}}
=> [6,2,3,4,5,1] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => 6
[7]
=> [[1,2,3,4,5,6,7]]
=> {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> {{1,3,4,5,6,7},{2}}
=> [3,2,4,5,6,7,1] => 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> {{1,4,5,6,7},{2},{3}}
=> [4,2,3,5,6,7,1] => 2
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> {{1,5,6,7},{2},{3},{4}}
=> [5,2,3,4,6,7,1] => 3
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> {{1,6,7},{2},{3},{4},{5}}
=> [6,2,3,4,5,7,1] => 4
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> {{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => 7
[8]
=> [[1,2,3,4,5,6,7,8]]
=> {{1,2,3,4,5,6,7,8}}
=> [2,3,4,5,6,7,8,1] => 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> {{1,3,4,5,6,7,8},{2}}
=> [3,2,4,5,6,7,8,1] => 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> {{1,4,5,6,7,8},{2},{3}}
=> [4,2,3,5,6,7,8,1] => 2
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> {{1,5,6,7,8},{2},{3},{4}}
=> [5,2,3,4,6,7,8,1] => 3
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> [2,3,4,1,6,7,8,5] => 2
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> [2,1,4,3,6,5,8,7] => 4
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,2,3,4,5,6,7,8] => 8
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> {{1,2,3,4,5,6,7,8,9}}
=> [2,3,4,5,6,7,8,9,1] => 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> {{1,3,4,5,6,7,8,9},{2}}
=> [3,2,4,5,6,7,8,9,1] => 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> {{1,4,5,6,7,8,9},{2},{3}}
=> [4,2,3,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> [1,2,3,4,5,6,7,8,9] => 9
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> {{1,2,3,4,5,6,7,8,9,10}}
=> [2,3,4,5,6,7,8,9,10,1] => 1
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> {{1,3,4,5,6,7,8,9,10},{2}}
=> [3,2,4,5,6,7,8,9,10,1] => 1
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> {{1,2},{3,4},{5,6},{7,8},{9,10}}
=> [2,1,4,3,6,5,8,7,10,9] => 5
[1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [1,2,3,4,5,6,7,8,9,10] => 10
Description
The number of adjacent cycles of a permutation.
This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).
Matching statistic: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1] => 1
[2]
=> [[1,2]]
=> [2] => [1] => 1
[1,1]
=> [[1],[2]]
=> [1,1] => [2] => 2
[3]
=> [[1,2,3]]
=> [3] => [1] => 1
[2,1]
=> [[1,3],[2]]
=> [1,2] => [1,1] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => [3] => 3
[4]
=> [[1,2,3,4]]
=> [4] => [1] => 1
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => [1,1] => 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => [2] => 2
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => [2,1] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [4] => 4
[5]
=> [[1,2,3,4,5]]
=> [5] => [1] => 1
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => [1,1] => 1
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => [1,1] => 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => [2,1] => 2
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => [1,2] => 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => [3,1] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [5] => 5
[6]
=> [[1,2,3,4,5,6]]
=> [6] => [1] => 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => [1,1] => 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => [1,1] => 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => [2,1] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => [2] => 2
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => [1,1,1] => 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => [3,1] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => [3] => 3
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => [2,2] => 2
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => [4,1] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => [6] => 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => [1] => 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [1,6] => [1,1] => 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [1,1,5] => [2,1] => 2
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [1,1,1,4] => [3,1] => 3
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [1,1,1,1,3] => [4,1] => 4
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => [7] => 7
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => [1] => 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [1,7] => [1,1] => 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [1,1,6] => [2,1] => 2
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [1,1,1,5] => [3,1] => 3
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [4,4] => [2] => 2
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [2,2,2,2] => [4] => 4
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [1,1,1,1,1,1,1,1] => [8] => 8
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [9] => [1] => 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [1,8] => [1,1] => 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [1,1,7] => [2,1] => 2
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [1,1,1,1,1,1,1,1,1] => [9] => 9
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [10] => [1] => 1
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [1,9] => [1,1] => 1
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => [5] => 5
[1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [1,1,1,1,1,1,1,1,1,1] => [10] => 10
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 70% ●values known / values provided: 94%●distinct values known / distinct values provided: 70%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 70% ●values known / values provided: 94%●distinct values known / distinct values provided: 70%
Values
[1]
=> [[1]]
=> [1] => [1] => 1
[2]
=> [[1,2]]
=> [2] => [1] => 1
[1,1]
=> [[1],[2]]
=> [1,1] => [2] => 2
[3]
=> [[1,2,3]]
=> [3] => [1] => 1
[2,1]
=> [[1,2],[3]]
=> [2,1] => [1,1] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => [3] => 3
[4]
=> [[1,2,3,4]]
=> [4] => [1] => 1
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => [1,1] => 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => [2] => 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => [1,2] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [4] => 4
[5]
=> [[1,2,3,4,5]]
=> [5] => [1] => 1
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => [1,1] => 1
[3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => [1,1] => 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => [1,2] => 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => [2,1] => 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,3] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [5] => 5
[6]
=> [[1,2,3,4,5,6]]
=> [6] => [1] => 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => [1,1] => 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => [1,1] => 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => [1,2] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => [2] => 2
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => [1,1,1] => 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => [1,3] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => [3] => 3
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => [2,2] => 2
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => [1,4] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => [6] => 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => [1] => 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [6,1] => [1,1] => 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [5,1,1] => [1,2] => 2
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => [1,3] => 3
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => [1,4] => 4
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => [7] => 7
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => [1] => 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [7,1] => [1,1] => 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => [1,2] => 2
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [5,1,1,1] => [1,3] => 3
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [4,4] => [2] => 2
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [2,2,2,2] => [4] => 4
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [1,1,1,1,1,1,1,1] => [8] => ? = 8
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [9] => [1] => 1
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [8,1] => [1,1] => 1
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [7,1,1] => [1,2] => 2
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [1,1,1,1,1,1,1,1,1] => [9] => ? = 9
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [10] => [1] => 1
[9,1]
=> [[1,2,3,4,5,6,7,8,9],[10]]
=> [9,1] => [1,1] => 1
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => [5] => 5
[1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [1,1,1,1,1,1,1,1,1,1] => [10] => ? = 10
Description
The last part of an integer composition.
Matching statistic: St000899
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000899: Integer compositions ⟶ ℤResult quality: 90% ●values known / values provided: 92%●distinct values known / distinct values provided: 90%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000899: Integer compositions ⟶ ℤResult quality: 90% ●values known / values provided: 92%●distinct values known / distinct values provided: 90%
Values
[1]
=> [[1]]
=> [1] => [1] => 1
[2]
=> [[1,2]]
=> [2] => [2] => 1
[1,1]
=> [[1],[2]]
=> [1,1] => [1,1] => 2
[3]
=> [[1,2,3]]
=> [3] => [3] => 1
[2,1]
=> [[1,3],[2]]
=> [1,2] => [2,1] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => [1,1,1] => 3
[4]
=> [[1,2,3,4]]
=> [4] => [4] => 1
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => [3,1] => 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => [2,2] => 2
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => [2,1,1] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [1,1,1,1] => 4
[5]
=> [[1,2,3,4,5]]
=> [5] => [5] => 1
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => [4,1] => 1
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => [3,2] => 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => [3,1,1] => 2
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => [2,1,2] => 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => [2,1,1,1] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,1,1,1,1] => 5
[6]
=> [[1,2,3,4,5,6]]
=> [6] => [6] => 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => [5,1] => 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => [4,2] => 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => [4,1,1] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => [3,3] => 2
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => [3,1,2] => 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => [3,1,1,1] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => [2,2,2] => 3
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => [2,1,1,2] => 2
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => [2,1,1,1,1] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => [1,1,1,1,1,1] => 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => [7] => 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [1,6] => [6,1] => 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [1,1,5] => [5,1,1] => 2
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [1,1,1,4] => [4,1,1,1] => 3
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [1,1,1,1,3] => [3,1,1,1,1] => 4
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 7
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => [8] => 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [1,7] => [7,1] => 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [1,1,6] => [6,1,1] => 2
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [1,1,1,5] => [5,1,1,1] => 3
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [4,4] => [4,4] => 2
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [2,2,2,2] => [2,2,2,2] => 4
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 8
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [9] => [9] => 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [1,8] => [8,1] => 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [1,1,7] => [7,1,1] => 2
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 9
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [10] => [10] => ? = 1
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [1,9] => [9,1] => ? = 1
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [2,2,2,2,2] => [2,2,2,2,2] => ? = 5
[1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => ? = 10
Description
The maximal number of repetitions of an integer composition.
This is the maximal part of the composition obtained by applying the delta morphism.
Matching statistic: St000326
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 90%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 90%
Values
[1]
=> [1]
=> 10 => 11 => 1
[2]
=> [1,1]
=> 110 => 111 => 1
[1,1]
=> [2]
=> 100 => 011 => 2
[3]
=> [1,1,1]
=> 1110 => 1111 => 1
[2,1]
=> [2,1]
=> 1010 => 1101 => 1
[1,1,1]
=> [3]
=> 1000 => 0011 => 3
[4]
=> [1,1,1,1]
=> 11110 => 11111 => 1
[3,1]
=> [2,1,1]
=> 10110 => 11011 => 1
[2,2]
=> [2,2]
=> 1100 => 0111 => 2
[2,1,1]
=> [3,1]
=> 10010 => 01101 => 2
[1,1,1,1]
=> [4]
=> 10000 => 00011 => 4
[5]
=> [1,1,1,1,1]
=> 111110 => 111111 => 1
[4,1]
=> [2,1,1,1]
=> 101110 => 110111 => 1
[3,2]
=> [2,2,1]
=> 11010 => 11101 => 1
[3,1,1]
=> [3,1,1]
=> 100110 => 011011 => 2
[2,2,1]
=> [3,2]
=> 10100 => 11001 => 1
[2,1,1,1]
=> [4,1]
=> 100010 => 001101 => 3
[1,1,1,1,1]
=> [5]
=> 100000 => 000011 => 5
[6]
=> [1,1,1,1,1,1]
=> 1111110 => 1111111 => 1
[5,1]
=> [2,1,1,1,1]
=> 1011110 => 1101111 => 1
[4,2]
=> [2,2,1,1]
=> 110110 => 111011 => 1
[4,1,1]
=> [3,1,1,1]
=> 1001110 => 0110111 => 2
[3,3]
=> [2,2,2]
=> 11100 => 01111 => 2
[3,2,1]
=> [3,2,1]
=> 101010 => 110101 => 1
[3,1,1,1]
=> [4,1,1]
=> 1000110 => 0011011 => 3
[2,2,2]
=> [3,3]
=> 11000 => 00111 => 3
[2,2,1,1]
=> [4,2]
=> 100100 => 011001 => 2
[2,1,1,1,1]
=> [5,1]
=> 1000010 => 0001101 => 4
[1,1,1,1,1,1]
=> [6]
=> 1000000 => 0000011 => 6
[7]
=> [1,1,1,1,1,1,1]
=> 11111110 => 11111111 => 1
[6,1]
=> [2,1,1,1,1,1]
=> 10111110 => 11011111 => 1
[5,1,1]
=> [3,1,1,1,1]
=> 10011110 => 01101111 => 2
[4,1,1,1]
=> [4,1,1,1]
=> 10001110 => 00110111 => 3
[3,1,1,1,1]
=> [5,1,1]
=> 10000110 => 00011011 => 4
[1,1,1,1,1,1,1]
=> [7]
=> 10000000 => 00000011 => 7
[8]
=> [1,1,1,1,1,1,1,1]
=> 111111110 => 111111111 => 1
[7,1]
=> [2,1,1,1,1,1,1]
=> 101111110 => 110111111 => 1
[6,1,1]
=> [3,1,1,1,1,1]
=> 100111110 => 011011111 => 2
[5,1,1,1]
=> [4,1,1,1,1]
=> 100011110 => 001101111 => 3
[4,4]
=> [2,2,2,2]
=> 111100 => 011111 => 2
[2,2,2,2]
=> [4,4]
=> 110000 => 000111 => 4
[1,1,1,1,1,1,1,1]
=> [8]
=> 100000000 => 000000011 => 8
[9]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => 1111111111 => ? = 1
[8,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => 1101111111 => ? = 1
[7,1,1]
=> [3,1,1,1,1,1,1]
=> 1001111110 => 0110111111 => 2
[1,1,1,1,1,1,1,1,1]
=> [9]
=> 1000000000 => 0000000011 => 9
[10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 11111111111 => ? = 1
[9,1]
=> [2,1,1,1,1,1,1,1,1]
=> 10111111110 => 11011111111 => ? = 1
[2,2,2,2,2]
=> [5,5]
=> 1100000 => 0000111 => 5
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> 10000000000 => 00000000011 => ? = 10
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000731
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [.,[.,.]]
=> [2,1] => 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> [1,3,2] => 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [2,3,1] => 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => 0 = 1 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2 = 3 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0 = 1 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0 = 1 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1 = 2 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 3 = 4 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[.,.],.],.],.],[.,.]]
=> [1,2,3,4,6,5] => 0 = 1 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 0 = 1 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 0 = 1 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 1 = 2 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 0 = 1 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 2 = 3 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> [2,3,4,5,6,1] => 4 = 5 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[[[[[.,.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,7,6] => 0 = 1 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[[[.,[.,.]],.],.],[.,.]]
=> [2,1,3,4,6,5] => 0 = 1 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0 = 1 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 1 = 2 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 1 = 2 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 2 = 3 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 2 = 3 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1 = 2 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[[[[.,[.,.]],.],.],.]]
=> [3,2,4,5,6,1] => 3 = 4 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => 5 = 6 - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,6,8,7] => 0 = 1 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[[[[.,[.,.]],.],.],.],[.,.]]
=> [2,1,3,4,5,7,6] => ? = 1 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[[.,[[.,.],.]],.],[.,.]]
=> [2,3,1,4,6,5] => 1 = 2 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 2 = 3 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[[[[.,.],[.,.]],.],.]]
=> [2,4,3,5,6,1] => 3 = 4 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => 6 = 7 - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,6,7,9,8] => 0 = 1 - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [2,1,3,4,5,6,8,7] => 0 = 1 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[[[.,[[.,.],.]],.],.],[.,.]]
=> [2,3,1,4,5,7,6] => ? = 2 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[.,[[[.,.],.],.]],[.,.]]
=> [2,3,4,1,6,5] => 2 = 3 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> [1,2,3,5,6,4] => 1 = 2 - 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,.],[[[[.,.],.],.],.]]
=> [1,3,4,5,6,2] => 3 = 4 - 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,1] => 7 = 8 - 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,6,7,8,10,9] => 0 = 1 - 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [[[[[[[.,[.,.]],.],.],.],.],.],[.,.]]
=> [2,1,3,4,5,6,7,9,8] => ? = 1 - 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [[[[[.,[[.,.],.]],.],.],.],[.,.]]
=> [2,3,1,4,5,6,8,7] => ? = 2 - 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,10,1] => 8 = 9 - 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,6,7,8,9,11,10] => 0 = 1 - 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
=> [2,1,3,4,5,6,7,8,10,9] => ? = 1 - 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => 4 = 5 - 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,10,11,1] => 9 = 10 - 1
Description
The number of double exceedences of a permutation.
A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St000877
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 90%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 90%
Values
[1]
=> [1]
=> 10 => 11 => 0 = 1 - 1
[2]
=> [1,1]
=> 110 => 111 => 0 = 1 - 1
[1,1]
=> [2]
=> 100 => 011 => 1 = 2 - 1
[3]
=> [1,1,1]
=> 1110 => 1111 => 0 = 1 - 1
[2,1]
=> [2,1]
=> 1010 => 1101 => 0 = 1 - 1
[1,1,1]
=> [3]
=> 1000 => 0011 => 2 = 3 - 1
[4]
=> [1,1,1,1]
=> 11110 => 11111 => 0 = 1 - 1
[3,1]
=> [2,1,1]
=> 10110 => 11011 => 0 = 1 - 1
[2,2]
=> [2,2]
=> 1100 => 0111 => 1 = 2 - 1
[2,1,1]
=> [3,1]
=> 10010 => 01101 => 1 = 2 - 1
[1,1,1,1]
=> [4]
=> 10000 => 00011 => 3 = 4 - 1
[5]
=> [1,1,1,1,1]
=> 111110 => 111111 => 0 = 1 - 1
[4,1]
=> [2,1,1,1]
=> 101110 => 110111 => 0 = 1 - 1
[3,2]
=> [2,2,1]
=> 11010 => 11101 => 0 = 1 - 1
[3,1,1]
=> [3,1,1]
=> 100110 => 011011 => 1 = 2 - 1
[2,2,1]
=> [3,2]
=> 10100 => 11001 => 0 = 1 - 1
[2,1,1,1]
=> [4,1]
=> 100010 => 001101 => 2 = 3 - 1
[1,1,1,1,1]
=> [5]
=> 100000 => 000011 => 4 = 5 - 1
[6]
=> [1,1,1,1,1,1]
=> 1111110 => 1111111 => 0 = 1 - 1
[5,1]
=> [2,1,1,1,1]
=> 1011110 => 1101111 => 0 = 1 - 1
[4,2]
=> [2,2,1,1]
=> 110110 => 111011 => 0 = 1 - 1
[4,1,1]
=> [3,1,1,1]
=> 1001110 => 0110111 => 1 = 2 - 1
[3,3]
=> [2,2,2]
=> 11100 => 01111 => 1 = 2 - 1
[3,2,1]
=> [3,2,1]
=> 101010 => 110101 => 0 = 1 - 1
[3,1,1,1]
=> [4,1,1]
=> 1000110 => 0011011 => 2 = 3 - 1
[2,2,2]
=> [3,3]
=> 11000 => 00111 => 2 = 3 - 1
[2,2,1,1]
=> [4,2]
=> 100100 => 011001 => 1 = 2 - 1
[2,1,1,1,1]
=> [5,1]
=> 1000010 => 0001101 => 3 = 4 - 1
[1,1,1,1,1,1]
=> [6]
=> 1000000 => 0000011 => 5 = 6 - 1
[7]
=> [1,1,1,1,1,1,1]
=> 11111110 => 11111111 => 0 = 1 - 1
[6,1]
=> [2,1,1,1,1,1]
=> 10111110 => 11011111 => 0 = 1 - 1
[5,1,1]
=> [3,1,1,1,1]
=> 10011110 => 01101111 => 1 = 2 - 1
[4,1,1,1]
=> [4,1,1,1]
=> 10001110 => 00110111 => 2 = 3 - 1
[3,1,1,1,1]
=> [5,1,1]
=> 10000110 => 00011011 => 3 = 4 - 1
[1,1,1,1,1,1,1]
=> [7]
=> 10000000 => 00000011 => 6 = 7 - 1
[8]
=> [1,1,1,1,1,1,1,1]
=> 111111110 => 111111111 => 0 = 1 - 1
[7,1]
=> [2,1,1,1,1,1,1]
=> 101111110 => 110111111 => 0 = 1 - 1
[6,1,1]
=> [3,1,1,1,1,1]
=> 100111110 => 011011111 => 1 = 2 - 1
[5,1,1,1]
=> [4,1,1,1,1]
=> 100011110 => 001101111 => 2 = 3 - 1
[4,4]
=> [2,2,2,2]
=> 111100 => 011111 => 1 = 2 - 1
[2,2,2,2]
=> [4,4]
=> 110000 => 000111 => 3 = 4 - 1
[1,1,1,1,1,1,1,1]
=> [8]
=> 100000000 => 000000011 => 7 = 8 - 1
[9]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => 1111111111 => ? = 1 - 1
[8,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => 1101111111 => ? = 1 - 1
[7,1,1]
=> [3,1,1,1,1,1,1]
=> 1001111110 => 0110111111 => 1 = 2 - 1
[1,1,1,1,1,1,1,1,1]
=> [9]
=> 1000000000 => 0000000011 => 8 = 9 - 1
[10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 11111111111 => ? = 1 - 1
[9,1]
=> [2,1,1,1,1,1,1,1,1]
=> 10111111110 => 11011111111 => ? = 1 - 1
[2,2,2,2,2]
=> [5,5]
=> 1100000 => 0000111 => 4 = 5 - 1
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> 10000000000 => 00000000011 => ? = 10 - 1
Description
The depth of the binary word interpreted as a path.
This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2].
The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].
Matching statistic: St000907
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 70% ●values known / values provided: 78%●distinct values known / distinct values provided: 70%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 70% ●values known / values provided: 78%●distinct values known / distinct values provided: 70%
Values
[1]
=> [1,0]
=> [1,0]
=> ([],1)
=> 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ([],2)
=> 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> ([(1,2)],3)
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([],6)
=> 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 3
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 4
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ([],7)
=> 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
=> 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ([(0,6),(6,1),(6,2),(6,3),(6,4),(6,5)],7)
=> 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ([(0,5),(5,6),(6,1),(6,2),(6,3),(6,4)],7)
=> 3
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> 4
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ([],8)
=> 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ? = 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ([(0,7),(7,1),(7,2),(7,3),(7,4),(7,5),(7,6)],8)
=> ? = 2
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ([(0,6),(6,7),(7,1),(7,2),(7,3),(7,4),(7,5)],8)
=> ? = 3
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 4
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ([],9)
=> ? = 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8)],9)
=> ? = 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ([(0,8),(8,1),(8,2),(8,3),(8,4),(8,5),(8,6),(8,7)],9)
=> ? = 2
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ([],10)
=> ? = 1
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9)],10)
=> ? = 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> 5
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 10
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St001483
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 76%●distinct values known / distinct values provided: 60%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 76%●distinct values known / distinct values provided: 60%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 4
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
Description
The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.
The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000648The number of 2-excedences of a permutation. St000686The finitistic dominant dimension of a Dyck path. St000039The number of crossings of a permutation. St000485The length of the longest cycle of a permutation. St000910The number of maximal chains of minimal length in a poset. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001933The largest multiplicity of a part in an integer partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000993The multiplicity of the largest part of an integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000454The largest eigenvalue of a graph if it is integral. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001330The hat guessing number of a graph. St001645The pebbling number of a connected graph. St001556The number of inversions of the third entry of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000007The number of saliances of the permutation. St001487The number of inner corners of a skew partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
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